A long known and studied approach, or `trajectory', to the SM is a linearly realized Effective Field Theory (SMEFT), see [1]} for a review, this road being pointed at by the decoupling theorem [2]}. The integration of any heavy particle whose mass can be arbitrarily larger than the EWSB vev (\(M>v\) ) in a perturbative linear realization will yield the SMEFT; supersymmetry or composite Higgs models fall into this category.
Is this the only road to the Standard Model, i.e. are there other consistent limits to obtain the SM couplings for the known spectrum of elementary particles? As fundamental as this topic is, on its present formulation the candidate preceded the question;
Higgs Effective Theory [3]}, [4]} is an EFT that encompasses the SMEFT but extends beyond it and might offer new roads. In HEFT, a linear realization is not assumed (though admissible in certain limit) and is indeed the most general Lorentz and gauge invariant theory with the known spectrum of particles (which suggests it should be possible to formulate it in terms of amplitudes). The theories that this EFT describes but fall out of SMEFT, which will be called here theories of the quotient space HEFT/SMEFT or simply quotient EFTs In other works these are called, with a slight abuse of notation, HEFT., could contain a path to the SM other than via SMEFT. This quotient space is characterized as missing a point in field space which is left invariant under an \(O(4)\)  transformation [5]}, [6]}, be it because it is not present or because the would be invariant point is singular [7]}. A geometric formalism was used to derive this result and also aids in exploring the properties of theories without field redundancies, as introduced in [5]}, [6]}, and followed up in [7]}, [11]}, [12]} - it is also adopted here.
Some theories in HEFT/SMEFT quotient space have been formulated while having a perturbative expansion [7]}; they have been found to have a cut-off of \(\sim 4 \pi v\)  and no limit can be taken within them that yields the SM. It has been suggested that all of this this quotient space shares this property of a finite \(v\) -bound cut-off  [14]} with further evidence provided in [12]}, which means in turn that they all could be casualties of our exploration with present and future machines. This question has been explored so far with perturbative unitarity bounds, while here it is looked at with semi-classical arguments.
